The administration estimates that fewer than 500,000 people have still not found coverage after their existing plans were canceled because of the new law’s requirement that insurance options meet a set of minimum requirements. Without this new exemption, those people would have faced a fine under the individual mandate, the provision of the ACA that requires people to purchase coverage, which is a core feature of the law.

To protect those who may be struggling to find replacement plans, Health and Human Services secretary Kathleen Sebelius said she had introduced a “temporary hardship exemption” that will enable those affected to either go without insurance or purchase only threadbare coverage without being penalised. (Bare-bones so-called “catastrophic plans”, which offer only minimal coverage, are ordinarily only permitted for people under the age of 30.)

Sebelius’ announcement, made late Thursday, came just days before the deadline for signing up for plans that go into effect on 1 January, and was immediately criticised by insurance companies, which could potentially lose customers who would otherwise have been compelled to sign up for their plans.

“This latest rule change could cause significant instability in the marketplace and lead to further confusion and disruption for consumers," said Robert Zirkelbach, spokesman for America's Health Insurance Plans.

On Wednesday, the industry extended the period in which it is allowing customers to pay their premiums – giving them an extra 10 days in January to complete their transactions. The move was designed to recognise the difficulties that people have had signing up for new health care plans.

Sebelius explained the decision in a letter to a group of Democratic senators, led by Mark Warner, who had raised concerns about people who had lost their coverage facing penalties for not being able to find new plans.

She said offering the option of catastrophic coverage would aid the “smooth transition” to the new health care marketplace, and added that a dedicated phone number had been set-up to deal with people with cancelled plans.